PLEASE, EMERGENCY, details please

The numbers 1,2,3,4,5,6,7,8,9 are arranged in a list so that each number is either greater than all the numbers that come before it or is less than all the numbers that come before it. For example, 4,5,6,3,2,7,1,8,9 is one such list: notice that (for instance) the 6 is greater than all the numbers that come before it, and the 2 is less than all the numbers that come before it. How many such lists of the numbers 1,2,3,4,5,6,7,8,9 are possible? Be sure to include complete explanations with your answer, using complete sentences. Imagine you were going to show your solution to a classmate, and try to write your solution so that he or she could understand it without doing any extra work.

How many such lists of the numbers 1,2,3,4,5,6,7,8,9 are possible?

\(n=9\)

There are \(2^{n-1}=2^{9-1}=2^8 = 256\) lists possible.

See Pascal's Triangle: Sum in line \(n-1 = 9-1 = 8\)

\(= ( 1+8+28+56+70+56+28+8+1 ) = 2^8 = 256\)

Lists:

1. ( 1, 2, 3, 4, 5, 6, 7, 8, 9 ) \(\qquad ( 1 \text{ list start with } 1 )\)

2. ( 2, 1, 3, 4, 5, 6, 7, 8, 9 )
3. ( 2, 3, 1, 4, 5, 6, 7, 8, 9 )
4. ( 2, 3, 4, 1, 5, 6, 7, 8, 9 )
5. ( 2, 3, 4, 5, 1, 6, 7, 8, 9 )
6. ( 2, 3, 4, 5, 6, 1, 7, 8, 9 )
7. ( 2, 3, 4, 5, 6, 7, 1, 8, 9 )
8. ( 2, 3, 4, 5, 6, 7, 8, 1, 9 )
9. ( 2, 3, 4, 5, 6, 7, 8, 9, 1 ) \(\qquad ( 8 \text{ lists start with } 2 )\)

10. ( 3, 2, 1, 4, 5, 6, 7, 8, 9 )
11. ( 3, 2, 4, 1, 5, 6, 7, 8, 9 )
12. ( 3, 2, 4, 5, 1, 6, 7, 8, 9 )
13. ( 3, 2, 4, 5, 6, 1, 7, 8, 9 )
14. ( 3, 2, 4, 5, 6, 7, 1, 8, 9 )
15. ( 3, 2, 4, 5, 6, 7, 8, 1, 9 )
16. ( 3, 2, 4, 5, 6, 7, 8, 9, 1 )
17. ( 3, 4, 2, 1, 5, 6, 7, 8, 9 )
18. ( 3, 4, 2, 5, 1, 6, 7, 8, 9 )
19. ( 3, 4, 2, 5, 6, 1, 7, 8, 9 )
20. ( 3, 4, 2, 5, 6, 7, 1, 8, 9 )
21. ( 3, 4, 2, 5, 6, 7, 8, 1, 9 )
22. ( 3, 4, 2, 5, 6, 7, 8, 9, 1 )
23. ( 3, 4, 5, 2, 1, 6, 7, 8, 9 )
24. ( 3, 4, 5, 2, 6, 1, 7, 8, 9 )
25. ( 3, 4, 5, 2, 6, 7, 1, 8, 9 )
26. ( 3, 4, 5, 2, 6, 7, 8, 1, 9 )
27. ( 3, 4, 5, 2, 6, 7, 8, 9, 1 )
28. ( 3, 4, 5, 6, 2, 1, 7, 8, 9 )
29. ( 3, 4, 5, 6, 2, 7, 1, 8, 9 )
30. ( 3, 4, 5, 6, 2, 7, 8, 1, 9 )
31. ( 3, 4, 5, 6, 2, 7, 8, 9, 1 )
32. ( 3, 4, 5, 6, 7, 2, 1, 8, 9 )
33. ( 3, 4, 5, 6, 7, 2, 8, 1, 9 )
34. ( 3, 4, 5, 6, 7, 2, 8, 9, 1 )
35. ( 3, 4, 5, 6, 7, 8, 2, 1, 9 )
36. ( 3, 4, 5, 6, 7, 8, 2, 9, 1 )
37. ( 3, 4, 5, 6, 7, 8, 9, 2, 1 ) \(\qquad ( 28 \text{ lists start with } 3 )\)

38. ( 4, 3, 2, 1, 5, 6, 7, 8, 9 )
39. ( 4, 3, 2, 5, 1, 6, 7, 8, 9 )
40. ( 4, 3, 2, 5, 6, 1, 7, 8, 9 )
41. ( 4, 3, 2, 5, 6, 7, 1, 8, 9 )
42. ( 4, 3, 2, 5, 6, 7, 8, 1, 9 )
43. ( 4, 3, 2, 5, 6, 7, 8, 9, 1 )
44. ( 4, 3, 5, 2, 1, 6, 7, 8, 9 )
45. ( 4, 3, 5, 2, 6, 1, 7, 8, 9 )
46. ( 4, 3, 5, 2, 6, 7, 1, 8, 9 )
47. ( 4, 3, 5, 2, 6, 7, 8, 1, 9 )
48. ( 4, 3, 5, 2, 6, 7, 8, 9, 1 )
49. ( 4, 3, 5, 6, 2, 1, 7, 8, 9 )
50. ( 4, 3, 5, 6, 2, 7, 1, 8, 9 )
51. ( 4, 3, 5, 6, 2, 7, 8, 1, 9 )
52. ( 4, 3, 5, 6, 2, 7, 8, 9, 1 )
53. ( 4, 3, 5, 6, 7, 2, 1, 8, 9 )
54. ( 4, 3, 5, 6, 7, 2, 8, 1, 9 )
55. ( 4, 3, 5, 6, 7, 2, 8, 9, 1 )
56. ( 4, 3, 5, 6, 7, 8, 2, 1, 9 )
57. ( 4, 3, 5, 6, 7, 8, 2, 9, 1 )
58. ( 4, 3, 5, 6, 7, 8, 9, 2, 1 )
59. ( 4, 5, 3, 2, 1, 6, 7, 8, 9 )
60. ( 4, 5, 3, 2, 6, 1, 7, 8, 9 )
61. ( 4, 5, 3, 2, 6, 7, 1, 8, 9 )
62. ( 4, 5, 3, 2, 6, 7, 8, 1, 9 )
63. ( 4, 5, 3, 2, 6, 7, 8, 9, 1 )
64. ( 4, 5, 3, 6, 2, 1, 7, 8, 9 )
65. ( 4, 5, 3, 6, 2, 7, 1, 8, 9 )
66. ( 4, 5, 3, 6, 2, 7, 8, 1, 9 )
67. ( 4, 5, 3, 6, 2, 7, 8, 9, 1 )
68. ( 4, 5, 3, 6, 7, 2, 1, 8, 9 )
69. ( 4, 5, 3, 6, 7, 2, 8, 1, 9 )
70. ( 4, 5, 3, 6, 7, 2, 8, 9, 1 )
71. ( 4, 5, 3, 6, 7, 8, 2, 1, 9 )
72. ( 4, 5, 3, 6, 7, 8, 2, 9, 1 )
73. ( 4, 5, 3, 6, 7, 8, 9, 2, 1 )
74. ( 4, 5, 6, 3, 2, 1, 7, 8, 9 )
75. ( 4, 5, 6, 3, 2, 7, 1, 8, 9 )
76. ( 4, 5, 6, 3, 2, 7, 8, 1, 9 )
77. ( 4, 5, 6, 3, 2, 7, 8, 9, 1 )
78. ( 4, 5, 6, 3, 7, 2, 1, 8, 9 )
79. ( 4, 5, 6, 3, 7, 2, 8, 1, 9 )
80. ( 4, 5, 6, 3, 7, 2, 8, 9, 1 )
81. ( 4, 5, 6, 3, 7, 8, 2, 1, 9 )
82. ( 4, 5, 6, 3, 7, 8, 2, 9, 1 )
83. ( 4, 5, 6, 3, 7, 8, 9, 2, 1 )
84. ( 4, 5, 6, 7, 3, 2, 1, 8, 9 )
85. ( 4, 5, 6, 7, 3, 2, 8, 1, 9 )
86. ( 4, 5, 6, 7, 3, 2, 8, 9, 1 )
87. ( 4, 5, 6, 7, 3, 8, 2, 1, 9 )
88. ( 4, 5, 6, 7, 3, 8, 2, 9, 1 )
89. ( 4, 5, 6, 7, 3, 8, 9, 2, 1 )
90. ( 4, 5, 6, 7, 8, 3, 2, 1, 9 )
91. ( 4, 5, 6, 7, 8, 3, 2, 9, 1 )
92. ( 4, 5, 6, 7, 8, 3, 9, 2, 1 )
93. ( 4, 5, 6, 7, 8, 9, 3, 2, 1 ) \(\qquad ( 56 \text{ lists start with } 4 )\)

94. ( 5, 4, 3, 2, 1, 6, 7, 8, 9 )
95. ( 5, 4, 3, 2, 6, 1, 7, 8, 9 )
96. ( 5, 4, 3, 2, 6, 7, 1, 8, 9 )
97. ( 5, 4, 3, 2, 6, 7, 8, 1, 9 )
98. ( 5, 4, 3, 2, 6, 7, 8, 9, 1 )
99. ( 5, 4, 3, 6, 2, 1, 7, 8, 9 )
100. ( 5, 4, 3, 6, 2, 7, 1, 8, 9 )
101. ( 5, 4, 3, 6, 2, 7, 8, 1, 9 )
102. ( 5, 4, 3, 6, 2, 7, 8, 9, 1 )
103. ( 5, 4, 3, 6, 7, 2, 1, 8, 9 )
104. ( 5, 4, 3, 6, 7, 2, 8, 1, 9 )
105. ( 5, 4, 3, 6, 7, 2, 8, 9, 1 )
106. ( 5, 4, 3, 6, 7, 8, 2, 1, 9 )
107. ( 5, 4, 3, 6, 7, 8, 2, 9, 1 )
108. ( 5, 4, 3, 6, 7, 8, 9, 2, 1 )
109. ( 5, 4, 6, 3, 2, 1, 7, 8, 9 )
110. ( 5, 4, 6, 3, 2, 7, 1, 8, 9 )
111. ( 5, 4, 6, 3, 2, 7, 8, 1, 9 )
112. ( 5, 4, 6, 3, 2, 7, 8, 9, 1 )
113. ( 5, 4, 6, 3, 7, 2, 1, 8, 9 )
114. ( 5, 4, 6, 3, 7, 2, 8, 1, 9 )
115. ( 5, 4, 6, 3, 7, 2, 8, 9, 1 )
116. ( 5, 4, 6, 3, 7, 8, 2, 1, 9 )
117. ( 5, 4, 6, 3, 7, 8, 2, 9, 1 )
118. ( 5, 4, 6, 3, 7, 8, 9, 2, 1 )
119. ( 5, 4, 6, 7, 3, 2, 1, 8, 9 )
120. ( 5, 4, 6, 7, 3, 2, 8, 1, 9 )
121. ( 5, 4, 6, 7, 3, 2, 8, 9, 1 )
122. ( 5, 4, 6, 7, 3, 8, 2, 1, 9 )
123. ( 5, 4, 6, 7, 3, 8, 2, 9, 1 )
124. ( 5, 4, 6, 7, 3, 8, 9, 2, 1 )
125. ( 5, 4, 6, 7, 8, 3, 2, 1, 9 )
126. ( 5, 4, 6, 7, 8, 3, 2, 9, 1 )
127. ( 5, 4, 6, 7, 8, 3, 9, 2, 1 )
128. ( 5, 4, 6, 7, 8, 9, 3, 2, 1 )
129. ( 5, 6, 4, 3, 2, 1, 7, 8, 9 )
130. ( 5, 6, 4, 3, 2, 7, 1, 8, 9 )
131. ( 5, 6, 4, 3, 2, 7, 8, 1, 9 )
132. ( 5, 6, 4, 3, 2, 7, 8, 9, 1 )
133. ( 5, 6, 4, 3, 7, 2, 1, 8, 9 )
134. ( 5, 6, 4, 3, 7, 2, 8, 1, 9 )
135. ( 5, 6, 4, 3, 7, 2, 8, 9, 1 )
136. ( 5, 6, 4, 3, 7, 8, 2, 1, 9 )
137. ( 5, 6, 4, 3, 7, 8, 2, 9, 1 )
138. ( 5, 6, 4, 3, 7, 8, 9, 2, 1 )
139. ( 5, 6, 4, 7, 3, 2, 1, 8, 9 )
140. ( 5, 6, 4, 7, 3, 2, 8, 1, 9 )
141. ( 5, 6, 4, 7, 3, 2, 8, 9, 1 )
142. ( 5, 6, 4, 7, 3, 8, 2, 1, 9 )
143. ( 5, 6, 4, 7, 3, 8, 2, 9, 1 )
144. ( 5, 6, 4, 7, 3, 8, 9, 2, 1 )
145. ( 5, 6, 4, 7, 8, 3, 2, 1, 9 )
146. ( 5, 6, 4, 7, 8, 3, 2, 9, 1 )
147. ( 5, 6, 4, 7, 8, 3, 9, 2, 1 )
148. ( 5, 6, 4, 7, 8, 9, 3, 2, 1 )
149. ( 5, 6, 7, 4, 3, 2, 1, 8, 9 )
150. ( 5, 6, 7, 4, 3, 2, 8, 1, 9 )
151. ( 5, 6, 7, 4, 3, 2, 8, 9, 1 )
152. ( 5, 6, 7, 4, 3, 8, 2, 1, 9 )
153. ( 5, 6, 7, 4, 3, 8, 2, 9, 1 )
154. ( 5, 6, 7, 4, 3, 8, 9, 2, 1 )
155. ( 5, 6, 7, 4, 8, 3, 2, 1, 9 )
156. ( 5, 6, 7, 4, 8, 3, 2, 9, 1 )
157. ( 5, 6, 7, 4, 8, 3, 9, 2, 1 )
158. ( 5, 6, 7, 4, 8, 9, 3, 2, 1 )
159. ( 5, 6, 7, 8, 4, 3, 2, 1, 9 )
160. ( 5, 6, 7, 8, 4, 3, 2, 9, 1 )
161. ( 5, 6, 7, 8, 4, 3, 9, 2, 1 )
162. ( 5, 6, 7, 8, 4, 9, 3, 2, 1 )
163. ( 5, 6, 7, 8, 9, 4, 3, 2, 1 ) \( \qquad ( 70 \text{ lists start with } 5 )\)

164. ( 6, 5, 4, 3, 2, 1, 7, 8, 9 )
165. ( 6, 5, 4, 3, 2, 7, 1, 8, 9 )
166. ( 6, 5, 4, 3, 2, 7, 8, 1, 9 )
167. ( 6, 5, 4, 3, 2, 7, 8, 9, 1 )
168. ( 6, 5, 4, 3, 7, 2, 1, 8, 9 )
169. ( 6, 5, 4, 3, 7, 2, 8, 1, 9 )
170. ( 6, 5, 4, 3, 7, 2, 8, 9, 1 )
171. ( 6, 5, 4, 3, 7, 8, 2, 1, 9 )
172. ( 6, 5, 4, 3, 7, 8, 2, 9, 1 )
173. ( 6, 5, 4, 3, 7, 8, 9, 2, 1 )
174. ( 6, 5, 4, 7, 3, 2, 1, 8, 9 )
175. ( 6, 5, 4, 7, 3, 2, 8, 1, 9 )
176. ( 6, 5, 4, 7, 3, 2, 8, 9, 1 )
177. ( 6, 5, 4, 7, 3, 8, 2, 1, 9 )
178. ( 6, 5, 4, 7, 3, 8, 2, 9, 1 )
179. ( 6, 5, 4, 7, 3, 8, 9, 2, 1 )
180. ( 6, 5, 4, 7, 8, 3, 2, 1, 9 )
181. ( 6, 5, 4, 7, 8, 3, 2, 9, 1 )
182. ( 6, 5, 4, 7, 8, 3, 9, 2, 1 )
183. ( 6, 5, 4, 7, 8, 9, 3, 2, 1 )
184. ( 6, 5, 7, 4, 3, 2, 1, 8, 9 )
185. ( 6, 5, 7, 4, 3, 2, 8, 1, 9 )
186. ( 6, 5, 7, 4, 3, 2, 8, 9, 1 )
187. ( 6, 5, 7, 4, 3, 8, 2, 1, 9 )
188. ( 6, 5, 7, 4, 3, 8, 2, 9, 1 )
189. ( 6, 5, 7, 4, 3, 8, 9, 2, 1 )
190. ( 6, 5, 7, 4, 8, 3, 2, 1, 9 )
191. ( 6, 5, 7, 4, 8, 3, 2, 9, 1 )
192. ( 6, 5, 7, 4, 8, 3, 9, 2, 1 )
193. ( 6, 5, 7, 4, 8, 9, 3, 2, 1 )
194. ( 6, 5, 7, 8, 4, 3, 2, 1, 9 )
195. ( 6, 5, 7, 8, 4, 3, 2, 9, 1 )
196. ( 6, 5, 7, 8, 4, 3, 9, 2, 1 )
197. ( 6, 5, 7, 8, 4, 9, 3, 2, 1 )
198. ( 6, 5, 7, 8, 9, 4, 3, 2, 1 )
199. ( 6, 7, 5, 4, 3, 2, 1, 8, 9 )
200. ( 6, 7, 5, 4, 3, 2, 8, 1, 9 )
201. ( 6, 7, 5, 4, 3, 2, 8, 9, 1 )
202. ( 6, 7, 5, 4, 3, 8, 2, 1, 9 )
203. ( 6, 7, 5, 4, 3, 8, 2, 9, 1 )
204. ( 6, 7, 5, 4, 3, 8, 9, 2, 1 )
205. ( 6, 7, 5, 4, 8, 3, 2, 1, 9 )
206. ( 6, 7, 5, 4, 8, 3, 2, 9, 1 )
207. ( 6, 7, 5, 4, 8, 3, 9, 2, 1 )
208. ( 6, 7, 5, 4, 8, 9, 3, 2, 1 )
209. ( 6, 7, 5, 8, 4, 3, 2, 1, 9 )
210. ( 6, 7, 5, 8, 4, 3, 2, 9, 1 )
211. ( 6, 7, 5, 8, 4, 3, 9, 2, 1 )
212. ( 6, 7, 5, 8, 4, 9, 3, 2, 1 )
213. ( 6, 7, 5, 8, 9, 4, 3, 2, 1 )
214. ( 6, 7, 8, 5, 4, 3, 2, 1, 9 )
215. ( 6, 7, 8, 5, 4, 3, 2, 9, 1 )
216. ( 6, 7, 8, 5, 4, 3, 9, 2, 1 )
217. ( 6, 7, 8, 5, 4, 9, 3, 2, 1 )
218. ( 6, 7, 8, 5, 9, 4, 3, 2, 1 )
219. ( 6, 7, 8, 9, 5, 4, 3, 2, 1 ) \(\qquad ( 56 \text{ lists start with } 6 )\)

220. ( 7, 6, 5, 4, 3, 2, 1, 8, 9 )
221. ( 7, 6, 5, 4, 3, 2, 8, 1, 9 )
222. ( 7, 6, 5, 4, 3, 2, 8, 9, 1 )
223. ( 7, 6, 5, 4, 3, 8, 2, 1, 9 )
224. ( 7, 6, 5, 4, 3, 8, 2, 9, 1 )
225. ( 7, 6, 5, 4, 3, 8, 9, 2, 1 )
226. ( 7, 6, 5, 4, 8, 3, 2, 1, 9 )
227. ( 7, 6, 5, 4, 8, 3, 2, 9, 1 )
228. ( 7, 6, 5, 4, 8, 3, 9, 2, 1 )
229. ( 7, 6, 5, 4, 8, 9, 3, 2, 1 )
230. ( 7, 6, 5, 8, 4, 3, 2, 1, 9 )
231. ( 7, 6, 5, 8, 4, 3, 2, 9, 1 )
232. ( 7, 6, 5, 8, 4, 3, 9, 2, 1 )
233. ( 7, 6, 5, 8, 4, 9, 3, 2, 1 )
234. ( 7, 6, 5, 8, 9, 4, 3, 2, 1 )
235. ( 7, 6, 8, 5, 4, 3, 2, 1, 9 )
236. ( 7, 6, 8, 5, 4, 3, 2, 9, 1 )
237. ( 7, 6, 8, 5, 4, 3, 9, 2, 1 )
238. ( 7, 6, 8, 5, 4, 9, 3, 2, 1 )
239. ( 7, 6, 8, 5, 9, 4, 3, 2, 1 )
240. ( 7, 6, 8, 9, 5, 4, 3, 2, 1 )
241. ( 7, 8, 6, 5, 4, 3, 2, 1, 9 )
242. ( 7, 8, 6, 5, 4, 3, 2, 9, 1 )
243. ( 7, 8, 6, 5, 4, 3, 9, 2, 1 )
244. ( 7, 8, 6, 5, 4, 9, 3, 2, 1 )
245. ( 7, 8, 6, 5, 9, 4, 3, 2, 1 )
246. ( 7, 8, 6, 9, 5, 4, 3, 2, 1 )
247. ( 7, 8, 9, 6, 5, 4, 3, 2, 1 ) \(\qquad ( 28 \text{ lists start with } 7 )\)

248. ( 8, 7, 6, 5, 4, 3, 2, 1, 9 )
249. ( 8, 7, 6, 5, 4, 3, 2, 9, 1 )
250. ( 8, 7, 6, 5, 4, 3, 9, 2, 1 )
251. ( 8, 7, 6, 5, 4, 9, 3, 2, 1 )
252. ( 8, 7, 6, 5, 9, 4, 3, 2, 1 )
253. ( 8, 7, 6, 9, 5, 4, 3, 2, 1 )
254. ( 8, 7, 9, 6, 5, 4, 3, 2, 1 )
255. ( 8, 9, 7, 6, 5, 4, 3, 2, 1 ) \(\qquad ( 8 \text{ lists start with } 8 )\)

256. ( 9, 8, 7, 6, 5, 4, 3, 2, 1 ) \(\qquad ( 1 \text{ list start with } 9 )\)

Thanks Heureka,

I will have to get my brain into gear to follow you on that one :)  

Nothing new there I suppose   

I write this before Heureka post his answer. It take a long time to get a logon screen.

Anyway I look over Heureka’s answer and it will take some time to study it, but my answer still seem like it is correct so I am posting it.

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The question DOSE NOT essentially ask how many combinations you can place the numbers in.

The question asks how many combinations you can place the numbers in where the number is either greater than or less than the ones before it. The only way a number can’t be greater than or less than the numbers before it is if the number is repeated.

So the question is how many combinations can you make without repeating a number. 

That is 9*8*7*6*5*4*3*2*1 = 362,880 ways,